# Math Table

## Welcome!

Math Table is a seminar jointly run by the Harvard Mathematics department and undergraduate students. The purpose of Math Table is to provide an opportunity for undergraduates to be exposed to interesting mathematical topics, as well as to gain experience in communicating and teaching mathematics.

Talks take place in Science Center 507 every Tuesday at 6:00 PM. Talks are catered, with different kinds of food every week. We do our best to accommodate all dietary needs, so if you have any concerns please send us an email in advance (see "About" tab for contacts).

### Who can attend/give talks?

All Harvard undergraduate students are welcome to attend any Math Table talk and to sign up to give a talk. Talks come in a wide array of topics, background levels, and styles (see the "Resources" tab). The Math Table organizers (see "About" tab) are here to help you pick topics and develop your talk, so even if you aren't sure about what your topic is, you should come speak with us!

To sign up to give a talk, or if you have any questions about Math Table, please send an email to spaul@math.harvard.edu and/or dzemke@g.harvard.edu, or to any of our undergraduate coordinators (see "About" tab for contacts).

## Upcoming Talks

### Why do Math?

Speaker: Professor Martin Nowak (Harvard University, Departments of Mathematics and Biology)

Abstract: Cooperation means that one individual pays a cost for another to receive a benefit. Cooperation can be at variance with natural selection: Why should you help a competitor? Yet cooperation is abundant in nature and is an important component of evolutionary innovation. Cooperation can be seen as the master architect of evolution and as the third fundamental principle of evolution beside mutation and selection. I will present mathematical principles of cooperation.

See http://math.harvard.edu/whydomath for more details

## Recent Talks

### Error Correcting Codes

Speaker: Garrett Brown

Abstract:

Imagine you as a child have developed a secret code with a friend that you wish to transmit over a string telephone. The problem is, string telephones aren’t always quite clear, and your friend doesn’t always receive the message perfectly. To solve this problem, you would like to develop a system where the message itself has added information (e.g. repetition) that makes it easier for your friend to correct a mistake if it should occur. This is the essence of error correcting codes. Error correcting codes are a surprising application of linear algebra to solve a discrete problem. In particular, it is an example of why linear algebra in its most general form is developed with the set of scalars being a field rather than just R or C. This talk will assume only that the audience has an idea of vectors in R n, and matrices with real entries.

### Euler In Action

Speaker: William Dunham, Harvard University

Abstract: Bill Dunham will speak at the first Math Table of the semester on Tue, Jan 29 at 6pm in SC 507. He will present one of Euler's solutions to the Basel problem (1 + 1/4 + 1/9 + ... = ???) in a way that will require only knowledge of l'Hospital's Rule and complex numbers, and give a little bit of historical background.

### Breaking Multivariable

Speaker: Stepan Paul

Abstract: If you want to understand something well, you should know how to break it. Most of our theorems in multivariable calculus (Clairaut's, Fubini's, ...) come with some condition about "nice" functions. In this talk, we look at some "naughty" functions to see when these theorems fail.

### Harmonic Forms on Surfaces: A Visual Approach

Speaker: Bjoern Meutzel (Dartmouth)

Abstract: We develop a unifying geometric intuition that allows us to gain insight into the relationship between the geometry of a surface and its harmonic forms. This approach is then applied to surfaces with small simple closed geodesics to gain insight into the energy distribution of these forms on the surface.

### Games in New Contexts

Speaker: Kelsey Houston-Edwards (Olin College)

Abstract: We'll play some familiar games in some unfamiliar contexts and try to generate conjectures about the possible outcomes. Then we'll prove some of these conjectures using theorems from combinatorics and game theory. This exploration should be accessible to all, with options to dive deeper into the proofs or just play some games.

### Why Five Platonic Solids? (It's not as simple as you think!)

Speaker: Glen Whitney

Abstract: There are various things in math that you just know are true: among all shapes with a given surface area, the sphere encloses the most volume, a trefoil knot is really knotted, a continuous simple closed curve has an inside and an outside, etc. Yet when you try to prove them, you suddenly discover that you are sticking your toes into deep (and possibly shark-infested) waters. At this math table, I'll try to convince you that the familiar fact that there are just five regular solids is a fact of this particularly vexing character (and hopefully have a bit of fun along the way).

### Divide and Conquer and 1-Center Clustering

Speaker: Shyam Narayanan

Abstract: Divide and Conquer is a common technique in theoretical computer science and has many cool applications! I'll begin by talking briefly about algorithms and the divide and conquer technique, and will explain how this technique can be used for the well-known Merge Sort algorithm. I'll finally explain the 1-Center Clustering with Outliers problem and show how one can get a linear-time algorithm for it using divide and conquer techniques.

### The Goldbach Comet

Speaker: Oliver Knill

Abstract:

The Goldbach comet is the graph of the function which tells in how many ways an integer 2n can be written as a sum of two primes. While it is preposterous trying to prove that the function is positive (this is the Goldbach conjecture), one can investigate the comet statistically and experiment in other number systems. I'm personally convinced that EVERY mathematician has secretly worked on the problem, but of course few admit it as there is a serious danger to reach a high score on the ``Prime numbers crackpot index" of Chris Caldwell (https://primes.utm.edu/notes/crackpot.html) Having not much to lose in matters of reputation, I can tell about my own foolish attempts which started 35 years ago both trying to find a root of the comet as well as trying to prove that none exists. There will be also a bit of history. While serious mathematicians use heavy analytic number theory, I will explain a simple real analysis attempt which only uses single variable calculus and relates to the Riemann zeta function. There is no reason for excitement: the approach does not prove it but there are interesting patterns and opportunities to experiment with computer algebra systems.

### Playing the St. Petersburg Paradox

Speaker: Mark Kong

Abstract: The Strong Law of Large Numbers is a way to formalize the notion that "Expected value is what you expect to get on average in the long run." We will discuss an analog of this claim for the St. Petersburg Paradox, a game with infinite expected value. Time permitting, we will also mention an analog of the Weak Law of Large Numbers.

Panelists TBA

### Eliminating Aperiodicity in Tilings

Speaker: Daniel Kim

Abstract: Given a set of tiles, can you use them to cover the entire plane without any overlaps? The attempt to answer this question led to the discovery of various tilings exhibiting intricate structures, such as the Wang tilings, the Penrose tilings, and the Socolar–Taylor tiling. In this talk, I will try to eliminate these beautiful tilings and make everything periodic and orderly.

### Math Experience Round Table, Hosted by CREWS

Come on by to share and learn what Harvard students have done with math outside their classes, such as REUs, internships, and other research. Dinner at 6:00, panel starts at 6:20. Sponsored by CREWS (Clusters Engaging Womxn in Mathematics).

The idea is to have students share (five minutes) the who, what, when, and where of a mathematical experience that was outside of their Harvard coursework. It would be great to highlight a variety of experiences ranging from intensive semesters abroad to seminars attended at MIT to internships to REUs or other work. It will be helpful for other students to hear how people found out about these experiences and the nuts and bolts of what was involved in participating!

### Tuesday, Sep 18 (5:30pm, SC 232)

Speaker: Sebastian Vasey

Title: Hypercomputation

Abstract: David Hilbert's "Entscheidungsproblem" asks for an algorithm to prove or disprove any mathematical statement. The non-existence of such an algorithm was proven independently by Church and Turing. This is often interpreted as meaning that there is no streamlined method to prove something: it takes hard work and creativity... Or does it? What did Church and Turing precisely prove? Could we imagine a hypercomputer capable of performing infinitely-many steps in a finite time? Could we build such a machine, or at least come close? I will discuss these questions and more.

### Tuesday, Sep 11 (5:30pm, SC 232)

Speaker: Davis Lazowski

Title: Quiver Representations

Abstract: Quiver representations give a beautiful, pictorial way to encode algebraic data. In this talk, the speaker will introduce quiver representations and study simple examples. At the end, the speaker will briefly survey applications to Lie theory and algebraic topology. The talk assumes some knowledge of linear algebra.

### Tuesday, Sep 4 (5:30pm, SC 232)

Speaker: Cliff Taubes

Title: Mysteries of 4 Dimensions

Abstract: The classification of spaces of dimensions 1-3 and 5—∞ is well understood. Dimension 4 is not understood. In fact, there are no compelling conjectures as to what the answer should be. I hope to explain some of this in the talk.

At 5:00pm in the same room, Gabriel Goldberg will be giving an overview of his fall tutorial "Infinite Combinatorics".